Geodesic
Involutions in the algebra of upper triangular matrices over the ring of algebraic integers of quadratic fields
Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 85-111

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The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields. The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions. In the case where the field $P$ has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra $T_n(P)$ for any natural number $n$ was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic $2$ or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classes of involutions were found. This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell's equations is significantly used.
Mots-clés : involutions
Keywords: the algebra of upper triangular matrices, the ring of algebraic integers of quadratic fields. 
I. A. Kulguskin. Involutions in the algebra of upper triangular matrices over the ring of algebraic integers of quadratic fields. Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 85-111. http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a5/
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[1] Krylov, P. A., Norbosambuyev, Ts. D., “Automorphisms of formal matrix algebras”, Sib. mat. Journal, 59:5 (2018), 1116–1127 | MR | Zbl

[2] Krylov, P. A., Tuganbaev, A. A., “Groups of automorphisms of rings of formal matrices”, Results of Science and Technology, 164 (2019), 96–124

[3] Kulguskin, I. A., Tapkin, D. T., “Involutions in the algebra of upper triangular matrices”, News of universities. Mathematics, 2023, no. 6, 11–30 | MR

[4] Leng, S., Algebraic numbers, Mir, M., 1966, 224 pp.

[5] Changa, M. E., Elementary theory of Pell equations, MPSU, M., 2019, 36 pp.

[6] Albert A. A., Structure of algebras, Amer. Math. Soc. Colloquium Publ., 24, 1961, 210 pp. | MR

[7] Brusamarello R., Fornaroli E. Z., Santulo Jr. E. A., “Classication of involutions on incidence algebras”, Comm. Alg., 39 (2011), 1941–1955 | DOI | MR | Zbl

[8] Brusamarello R., Fornaroli E. Z., Santulo Jr. E. A., “Anti-automorphisms and involutions on (finitary) incidence algebras”, Linear Multilinear Algebra, 60 (2012), 181–188 | DOI | MR | Zbl

[9] Brusamarello R., Lewis D. W., “Automorphisms and involutions on incidence algebras”, Linear and Multilinear Algebra, 59:11 (2011), 1247–1267 | DOI | MR | Zbl

[10] Fornaroli E. Z., Pezzott R. E. M., “Anti-isomorphisms and involutions on the idealization of the incidence space over the finitary incidence algebra”, Linear Algebra Appl., 637 (2022), 82–109 | DOI | MR | Zbl

[11] Jacobson N., Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996, 284 pp. | MR | Zbl

[12] Knus M. A., Merkurjev A., Rost M., Tignol J.-P., The Book of Involutions, Amer. Math. Soc. Colloquium Publ., 44, 1998, 31 pp. | DOI | MR | Zbl

[13] Krylov P. A., Tuganbaev A. A., “Automorphisms of Formal Matrix Rings”, J. Math. Sci., 258:2 (2021), 222–249 | DOI | MR | Zbl

[14] Spiegel E., “Involutions in incidence algebras”, Linear Algebra App., 405 (2005), 155–162 | DOI | MR | Zbl

[15] Vincenzo O. M., Koshlukov P., Scala R., “Involutions for upper triangular matrix algebras”, Advances in Applied Mathematics, 37 (2006), 541–568 | DOI | MR | Zbl